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A110441 Triangular array formed by the Mersenne numbers. 4
1, 3, 1, 7, 6, 1, 15, 23, 9, 1, 31, 72, 48, 12, 1, 63, 201, 198, 82, 15, 1, 127, 522, 699, 420, 125, 18, 1, 255, 1291, 2223, 1795, 765, 177, 21, 1, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence factors A038255 into a product of Riordan arrays.

Subtriangle of the triangle given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

From Peter Bala, Jul 22 2014: (Start)

Let M denote the lower unit triangular array A130330 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/

having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

For 1<=k<=n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01 and 02. - Milan Janjic, Dec 20 2016

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, J. Int. Seq. 8 (2005), #05.3.7.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

FORMULA

Riordan array M(n, k): (1/(1-3z+2z^2), z/(1-3z+2z^2)). Leftmost column M(n, 0) is the Mersenne numbers A000225, first column is A045618, second column is A055582, row sum is A007070 and diagonal sum is even-indexed Fibonacci numbers A001906.

T(n,k) = Sum_{j=0..n} C(j+k,k)C(n-j,k)2^(n-j-k). - Paul Barry, Feb 13 2006

From Philippe Deléham, Mar 19 2012: (Start)

G.f.: 1/(1-(3+y)*x+2*x^2).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -2*T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000225(n+1), A007070(n), A107839(n), A154244(n), A186446(n), A190975(n+1), A190979(n+1), A190869(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7 respectively. (End)

Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (2^(i+1) - 1)*T(n-i,k). - Peter Bala, Jul 22 2014

EXAMPLE

Triangle starts:

   1;

   3,  1;

   7,  6,  1;

  15, 23,  9,  1;

  31, 72, 48, 12,  1;

(0, 3, -2/3, 2/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:

  1

  0,  1

  0,  3,  1

  0,  7,  6,  1

  0, 15, 23,  9,  1

  0, 31, 72, 48, 12, 1. - Philippe Deléham, Mar 19 2012

With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins

/ 1          \/1         \/1        \      / 1       \

| 3  1       ||0  1      ||0 1      |      | 3  1    |

| 7  3 1     ||0  3 1    ||0 0 1    |... = | 7  6 1  |

|15  7 3 1   ||0  7 3 1  ||0 0 3 1  |      |15 23 9 1|

|31 15 7 3 1 ||0 15 7 3 1||0 0 7 3 1|      |...      |

|...         ||...       ||...      |      |...      | - Peter Bala, Jul 22 2014

MATHEMATICA

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - (3 + y) x + 2 x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

CROSSREFS

Cf. A000225, A130330, A206306.

Sequence in context: A213043 A319740 A275662 * A111806 A321163 A054458

Adjacent sequences:  A110438 A110439 A110440 * A110442 A110443 A110444

KEYWORD

easy,nonn,tabl

AUTHOR

Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005

STATUS

approved

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Last modified January 21 17:07 EST 2019. Contains 319350 sequences. (Running on oeis4.)